Do probabilists flip coins all the time?

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1 Include Only If Paper Has a Subtitle Department of Mathematics and Statistics Do probabilists flip coins all the time? Math Graduate Seminar February 17, 2009

2 Outline 1 Bernoulli random variables and Rademacher functions 2 3

3 Marks D = { 0, 1 } S = { 1, 1 } : d = 1 + s 2 or s = 2d 1

4 Binary expansion x = n=1 Cantor set C: x n 2 n Bernoulli random variables: x n : [0, 1] { 0, 1 } c = n=1 c n 3 n, c n : C { 0, 2 }. (Both [0, 1] and C are probability spaces.)

5 Square bump and square wave { 0 if 0 < x < 1 r( ) = periodic extension of h( ), where h(x) = 1 if 1 < x < 2 (Choose values at the jumps as desired, e.g., to make the function left-continuous.) Now, squeeze and restrict to [0, 1], x n = x n (x) = r(2 n x), n = 1, 2,..., x [0, 1],

6 Rademacher and Haar functions Shift and scale the marks D S-square wave. Or, directly: ( ) ρ n (x) = sign sin(2 n π x), x [0, 1]. Haar function: a single double-tooth piece of a Rademacher function; a dyadic squeeze and shift of the square bump 2h(x) 1. 2 n of Haar functions of order n form a basis in the vector space D n of piecewise functions, constant on dyadic intervals ( k 1 2 n, k ] 2 n, k = 1,..., 2 n

7 Walsh functions Rademacher functions span only an n-dimensional subspace. Now, span the products, a.k.a. Walsh functions: w d1,...,d n = ρ d 1 1 ρdn n, d 1,..., d n { 0, 1 }. Since there are 2 n Walsh functions and they are independent, they also form a basis of D n.

8 Powers Walsh functions can be ordered lexicographically w 1 = w 1, w 2 = w 01, w 3 = w 11, w 4 = w 001, w 5 = w 101, w 6 = w 011,... (skipping zeros on the right). Indicate a sequence by a boldface font, e.g., d = (d n ). Put D = { d = (d n ) : d n = 0 eventually }. Define the power w d = ρ d, d D.

9 Rademacher chaos - bras and kets Walsh polynomials, a.k.a. Rademacher random chaos: x ρ = x d ρ d, d D The braket indicates the interaction between the array x = [x d ] of coefficients and the power-bearing sequence ρ. The symbol can be split into two parts, called by Paul Dirac bra : x and ket : ρ

10 Second order chaos In the Hilbert space L 2 [0, 1] of square-integrable functions with the norm ( 1/2 1 f 2 = f (t) dt) 2, the Walsh functions form an orthonormal basis. By the Pythagorean, a.k.a. Parseval s, theorem x 0 2 ρ = d D x d 2. That is, every f L 2 [0, 1] admits a Rademacher chaos representation.

11 First order chaos Now, let f L 1 [0, 1], merely integrable: f 1 = 1 0 f (t) dt <. The projections onto the dyadic subspaces of order n P n f = E [ f D n ] are well defined and converge to f in the norm and almost everywhere (this requires more math). They are also known as conditional expectations or martingales in probability theory. Again, every integrable function on [0, 1] admits a Rademacher chaos representation.

12 Derivatives In the classical calculus, for x = (x n ), D j x d = x j x d = 1 x j x d 1I {dj 0} This operator acts also on Rademacher powers: D j f = E [ f ρ j span { ρ i : i j } ] That is, D j ρ d removes the factor ρ j from the power when it is present, otherwise the result is zero. The sequence D = (D j ) yields a variety of differential operators a D = a c D c. c

13 Random walk Norbert Wiener (1930 s): polynomial random chaos is the algebraic mixture of sums of products of i.i.d. random variables. Wiener s purpose: to model fluid and gas dynamics. By the Central Limit Theorem (de Moivre, 17th century) ρ ρ n n is approximately Gauss. The refined and scaled random walk becomes the Brownian Motion B(t) in the limit. Rademacher random chaos entails the Gaussian chaos.

14 Random chaos Geometric Brownian Motion is a functional of BM: e αb(t) β t. Theorem Every reasonable functional of Brownian Motion admits a random chaos representation: x d γ d, d where γ = (γ n ) are i.i.d. standard normal random variables.